Classical Feedback Vertex Set Problem Research Articles

Preprocessing to reduce the search space: Antler structures for feedback vertex set

The goal of this paper is to open up a new research direction aimed at understanding the power of preprocessing in speeding up algorithms that solve NP-hard problems exactly. We explore this direction for the classic Feedback Vertex Set problem on undirected graphs, leading to a new type of graph structure called antler decomposition, which identifies vertices that belong to an optimal solution. It is an analogue of the celebrated crown decomposition which has been used for Vertex Cover. We develop the graph structure theory around such decompositions and develop fixed-parameter tractable algorithms to find them, parameterized by the number of vertices for which they witness presence in an optimal solution. This reduces the search space of fixed-parameter tractable algorithms parameterized by the solution size that solve Feedback Vertex Set.

A parameterized algorithm for subset feedback vertex set in tournaments

The Subset Feedback Vertex Set problem (SFVS) takes an n-vertex graph G=(V,E), a terminal set T⊆V, and an integer k as the input. The goal is to determine whether there exists a subset S⊆V of at most k vertices whose removal makes no terminal in T contained in a cycle in the remaining graph. When T=V, SFVS degenerates to the classical Feedback Vertex Set problem (FVS). Both SFVS and FVS have been extensively studied in parameterized algorithms. In this paper, we study parameterized algorithms for Subset Feedback Vertex Set in Tournaments (SFVST), i.e., SFVS with the restriction that the input graph is always a tournament. By using the iterative compression method and a novel dynamic programming, we show that SFVST can be solved in 2k+o(k)nO(1) time, improving the bound obtained from 3-Hitting Set.

Simultaneous Feedback Edge Set: A Parameterized Perspective

Agrawal et al. (ACM Trans Comput Theory 10(4):18:1–18:25, 2018. https://doi.org/10.1145/3265027 ) studied a simultaneous variant of the classic Feedback Vertex Set problem, called Simultaneous Feedback Vertex Set (Sim-FVS). Here, we consider the edge variant of the problem, namely, Simultaneous Feedback Edge Set (Sim-FES). In this problem, the input is an n-vertex graph G, a positive integer k, and a coloring function col: $$E(G) \rightarrow 2^{[\alpha ]}$$ , and the objective is to check whether there is an edge subset S of cardinality k in G such that for each $$i \in [\alpha ]$$ , $$G_i - S$$ is acyclic. Unlike the vertex variant of the problem, when $$\alpha =1$$ , the problem is equivalent to finding a maximal spanning forest and hence it is polynomial time solvable. We show that for $$\alpha =3$$ , Sim-FES is NP-hard, and does not admit an algorithm of running time $$2^{o(k)}n^{{{\mathcal {O}}}(1)}$$ unless ETH fails. This hardness result is complimented by an FPT algorithm for Sim-FES running in time $$2^{\omega k \alpha +\alpha \log k} n^{{{\mathcal {O}}}(1)}$$ where $$\omega$$ is the exponent in the running time of matrix multiplication. The same algorithm gives a polynomial time algorithm for the case when $$\alpha =2$$ . We also give a kernel for Sim-FES with $$(k\alpha )^{{\mathcal {O}}(\alpha )}$$ vertices. Finally, we consider a “dual” version of the problem called Maximum Simultaneous Acyclic Subgraph and give an FPT algorithm with running time $$2^{\omega q \alpha }n^{{\mathcal {O}}(1)}$$ , where q is the number of edges in the output subgraph.

An Improved FPT Algorithm for Independent Feedback Vertex Set

We study the Independent Feedback Vertex Set problem — a variant of the classic Feedback Vertex Set problem where, given a graph G and an integer k, the problem is to decide whether there exists a vertex set Ssubseteq V(G) such that G ∖ S is a forest and S is an independent set of size at most k. We present an mathcal {O}^{ast }((1+varphi ^{2})^{k})-time FPT algorithm for this problem, where φ < 1.619 is the golden ratio, improving the previous fastest mathcal {O}^{ast }(4.1481^{k})-time algorithm given by Agrawal et al. (2016). The exponential factor in our time complexity bound matches the fastest deterministic FPT algorithm for the classic Feedback Vertex Set problem. On the technical side, the main novelty is a refined measure of an input instance in a branching process, that allows for a simpler and more concise description and analysis of the algorithm.

Subset feedback vertex set on graphs of bounded independent set size

The (Weighted) Subset Feedback Vertex Set problem is a generalization of the classical Feedback Vertex Set problem and asks for a vertex set of minimum (weight) size that intersects all cycles containing a vertex of a predescribed set of vertices. Although Subset Feedback Vertex Set and Feedback Vertex Set exhibit different computational complexity on split graphs, no similar characterization is known on other classes of graphs. Towards the understanding of the complexity difference between the two problems, it is natural to study the importance of structural graph parameters. Here we consider graphs of bounded independent set number for which it is known that Weighted Feedback Vertex Set can be solved in polynomial time. We provide a dichotomy result with respect to the size α of a maximum independent set. In particular we show that Weighted Subset Feedback Vertex Set can be solved in polynomial time for graphs with α≤3, whereas we prove that the problem remains NP-hard for graphs with α≥4. Moreover, we show that the (unweighted) Subset Feedback Vertex Set problem can be solved in polynomial time on graphs of bounded independent set number by giving an algorithm with running time nO(α). To complement our results, we demonstrate how our ideas can be extended to other terminal set problems on graphs of bounded independent set size. Node Multiway Cut is a terminal set problem that asks for a vertex set of minimum size that intersects all paths connecting any two terminals. Based on our findings for Subset Feedback Vertex Set, we settle the complexity of Node Multiway Cut as well as its variants where nodes are weighted and/or the terminals are deletable, for every value of the given independent set number.

Polynomial-time algorithms for the subset feedback vertex set problem on interval graphs and permutation graphs

Given a vertex-weighted graph G=(V,E) and a set S⊆V, a subset feedback vertex set X is a set of the vertices of G such that the graph induced by V∖X has no cycle containing a vertex of S. The Subset Feedback Vertex Set problem takes as input G and S and asks for the subset feedback vertex set of minimum total weight. In contrast to the classical Feedback Vertex Set problem which is obtained from the Subset Feedback Vertex Set problem for S=V, restricted to graph classes the Subset Feedback Vertex Set problem is known to be NP-complete on split graphs and, consequently, on chordal graphs. However, whereas Feedback Vertex Set is polynomially solvable for AT-free graphs, no such result is known for the Subset Feedback Vertex Set problem on any subclass of AT-free graphs. Here we give the first polynomial-time algorithms for the problem on two subclasses of AT-free graphs: interval graphs and permutation graphs. Moreover, towards the unknown complexity of the problem for AT-free graphs, we give a polynomial-time algorithm for co-bipartite graphs. Thus, we contribute to the first positive results of the Subset Feedback Vertex Set problem when restricted to graph classes for which Feedback Vertex Set is solved in polynomial time.

Bivariate complexity analysis of Almost Forest Deletion

In this paper we study a generalization of classic Feedback Vertex Set problem in the realm of multivariate complexity analysis. We say that a graph F is an l-forest if we can delete at most l edges from F to get a forest. That is, F is at most l edges away from being a forest. In this paper we introduce the Almost Forest Deletion problem, where given a graph G and integers k and l, the question is whether there exists a subset of at most k vertices such that its deletion leaves us an l-forest. We show that this problem admits an algorithm with running time 2O(k+l)nO(1) and a kernel of size O(kl(k+l)). We also show that the problem admits a 2O(tw)nO(1) algorithm on bounded treewidth graphs, using which we design a subexponential algorithm for the problem on planar graphs.

A Randomized Polynomial Kernel for Subset Feedback Vertex Set

The subset feedback vertex set problem generalizes the classical feedback vertex set problem and asks, for a given undirected graph G = (V, E), a set S ⊆ V, and an integer k, whether there exists a set X of at most k vertices such that no cycle in G ź X contains a vertex of S. It was independently shown by Cygan et al. (ICALP '11, SIDMA '13) and Kawarabayashi and Kobayashi (JCTB '12) that subset feedback vertex set is fixed-parameter tractable for parameter k. Cygan et al. asked whether the problem also admits a polynomial kernelization. We answer the question of Cygan et al. positively by giving a randomized polynomial kernelization for the equivalent version where S is a set of edges. In a first step we show that edge subset feedback vertex set has a randomized polynomial kernel parameterized by |S| + k with źź(|S|2k)$\mathcal {O}(|S|^{2}k)$ vertices. For this we use the matroid-based tools of Kratsch and Wahlstrom (FOCS '12) that for example were used to obtain a polynomial kernel for s-multiway cut. Next we present a preprocessing that reduces the given instance (G, S, k) to an equivalent instance (Gź, Sź, kź) where the size of Sź is bounded by źź(k4)$\mathcal {O}(k^{4})$. These two results lead to a randomized polynomial kernel for subset feedback vertex set with źź(k9)$\mathcal {O}(k^{9})$ vertices.

On Group Feedback Vertex Set Parameterized by the Size of the Cutset

We study parameterized complexity of a generalization of the classical Feedback Vertex Set problem, namely the Group Feedback Vertex Set problem: we are given a graph $$G$$ with edges labeled with group elements, and the goal is to compute the smallest set of vertices that hits all cycles of $$G$$ that evaluate to a non-null element of the group. This problem generalizes not only Feedback Vertex Set, but also Subset Feedback Vertex Set, Multiway Cut and Odd Cycle Transversal . Completing the results of Guillemot (Discrete Optim 8(1):61–71, 2011), we provide a fixed-parameter algorithm for the parameterization by the size of the cutset only. Our algorithm works even if the group is given as a blackbox performing group operations.

Subset Feedback Vertex Set Is Fixed-Parameter Tractable

The classical Feedback Vertex Set problem asks, for a given undirected graph $G$ and an integer $k$, to find a set of at most $k$ vertices that hits all the cycles in the graph $G$. Feedback Vertex Set has attracted a large amount of research in the parameterized setting, and subsequent fixed-parameter and kernelization algorithms have been a rich source of ideas in the field. In this paper we consider a more general and difficult version of the problem, named Subset Feedback Vertex Set (Subset-FVS), where an instance comes additionally with a set $S \subseteq V$ of vertices, and we ask for a set of at most $k$ vertices that hits all simple cycles passing through $S$. Because of its applications in circuit testing and genetic linkage analysis, Subset-FVS was studied from the approximation algorithm perspective by Even et al. [SIAM J. Discrete Math., 13 (2000), pp. 225--267; SIAM J. Comput., 30 (2000), pp. 1231--1252]. The question of whether the Subset-FVS problem is fixed-parameter tractable was posed in...

Hardness of Vertex Deletion and Project Scheduling

Assuming the Unique Games Conjecture, we show strong inapproximability results for two natural vertex deletion problems on directed graphs: for any integer $k\geq 2$ and arbitrary small $\epsilon > 0$, the Feedback Vertex Set problem and the DAG Vertex Deletion problem are inapproximable within a factor $k-\epsilon$ even on graphs where the vertices can be almost partitioned into $k$ solutions. This gives a more structured and yet simpler (albeit using the “It Ain't Over Till It's Over” theorem) UG-hardness result for the Feedback Vertex Set problem than the previous hardness result. In comparison to the classical Feedback Vertex Set problem, the DAG Vertex Deletion problem has received little attention and, although we think it is a natural and interesting problem, the main motivation for our inapproximability result stems from its relationship with the classical Discrete Time-Cost Tradeoff Problem. More specifically, our results imply that the deadline version is UG-hard to approximate within any constant. This explains the difficulty in obtaining good approximation algorithms for that problem and further motivates previous alternative approaches such as bicriteria approximations. An extended abstract of this paper appeared in the Proceedings of the 15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, 2012 (APPROX'12).

On Parameterized Independent Feedback Vertex Set

We investigate a generalization of the classical Feedback Vertex Set (FVS) problem from the point of view of parameterized algorithms. Independent Feedback Vertex Set (IFVS) is the “independent” variant of the FVS problem and is defined as follows: given a graph G and an integer k, decide whether there exists F⊆V(G), |F|≤k, such that G[V(G)∖F] is a forest and G[F] is an independent set; the parameter is k. Note that the similarly parameterized version of the FVS problem–where there is no restriction on the graph G[F]–has been extensively studied in the literature. The connected variant CFVS–where G[F] is required to be connected–has received some attention as well. The FVS problem easily reduces to the IFVS problem in a manner that preserves the solution size, and so any algorithmic result for IFVS directly carries over to FVS. We show that IFVS can be solved in O(5knO(1)) time, where n is the number of vertices in the input graph G, and obtain a cubic (O(k3)) kernel for the problem. Note the contrast with the CFVS problem, which does not admit a polynomial kernel unless CoNP⊆NP/Poly.

FPT algorithms for Connected Feedback Vertex Set

We study the recently introduced Connected Feedback Vertex Set (CFVS) problem from the view-point of parameterized algorithms. CFVS is the connected variant of the classical Feedback Vertex Set problem and is defined as follows: given a graph G=(V,E) and an integer k, decide whether there exists F?V, |F|?k, such that G[V?F] is a forest and G[F] is connected. We show that Connected Feedback Vertex Set can be solved in time O(2 O(k) n O(1)) on general graphs and in time $O(2^{O(\sqrt{k}\log k)}n^{O(1)})$ on graphs excluding a fixed graph H as a minor. Our result on general undirected graphs uses, as a subroutine, a parameterized algorithm for Group Steiner Tree, a well studied variant of Steiner Tree. We find the algorithm for Group Steiner Tree of independent interest and believe that it could be useful for obtaining parameterized algorithms for other connectivity problems.